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# Problem 315: Digital root clocks

(see projecteuler.net/problem=315)

Sam and Max are asked to transform two digital clocks into two "digital root" clocks.

A digital root clock is a digital clock that calculates digital roots step by step.

When a clock is fed a number, it will show it and then it will start the calculation, showing all the intermediate values until it gets to the result.

For example, if the clock is fed the number 137, it will show: "137" → "11" → "2" and then it will go black, waiting for the next number.

Every digital number consists of some light segments: three horizontal (top, middle, bottom) and four vertical (top-left, top-right, bottom-left, bottom-right).

Number "1" is made of vertical top-right and bottom-right, number "4" is made by middle horizontal and vertical top-left, top-right and bottom-right. Number "8" lights them all.

The clocks consume energy only when segments are turned on/off.

To turn on a "2" will cost 5 transitions, while a "7" will cost only 4 transitions.

Sam and Max built two different clocks.

Sam's clock is fed e.g. number 137: the clock shows "137", then the panel is turned off, then the next number ("11") is turned on, then the panel is turned off again and finally the last number ("2") is turned on and, after some time, off.

For the example, with number 137, Sam's clock requires:

"137": (2 + 5 + 4) * 2 = 22 transitions ("137" on/off).

"11": (2 + 2) * 2 = 8 transitions ("11" on/off).

"2": (5) * 2 = 10 transitions ("2" on/off).

For a grand total of 40 transitions.

Max's clock works differently. Instead of turning off the whole panel, it is smart enough to turn off only those segments that won't be needed for the next number.

For number 137, Max's clock requires:

"137": 2 + 5 + 4 = 11 transitions ("137" on)

7 transitions (to turn off the segments that are not needed for number "11").

"11": 0 transitions (number "11" is already turned on correctly)

3 transitions (to turn off the first "1" and the bottom part of the second "1"; the top part is common with number "2").

"2": 4 transitions (to turn on the remaining segments in order to get a "2")

5 transitions (to turn off number "2").

For a grand total of 30 transitions.

Of course, Max's clock consumes less power than Sam's one. The two clocks are fed all the prime numbers between A = 10^7 and B = 2 * 10^7.

Find the difference between the total number of transitions needed by Sam's clock and that needed by Max's one.

# My Algorithm

I encode all lit segments as bitmasks in `Segments`

. Counting the set bits (`popcnt`

) is equivalent to counting the lit segments.

There are at most 8 digits, hence a 64 bit integer represents all segments of a number.

Sam's algorithm can be found in `sam`

: switching on and off all segments "costs" 2x `popcnt(getSegments(x))`

.

Max's algorithm (see `max`

) is a bit more complicated: I have to compare all lit segments of the previous number to the lit segments of the current number.

When I XOR both bitmasks then I get a new bitmasks with all segments that change (either have to be turned off or turnd on).

Counting the set bits in the XORed value tells me the number of transitions.

Before the first and after the last number all lit segments transition from and to the "empty" state.

Both routines work recursively because I hoped to improve the execution time by caching / memoizing results:

After the first step the `digitSum`

is below 70. However, the simple memoization in `sam`

didn't help much

and therefore I didn't even attempt to code something similiar in `max`

.

## Note

`popcnt`

was named after the super-fast CPU instruction which counts bits.

My G++ compiler shows the result after 0.12 seconds when that intrinsic can be used but needs 0.17 seconds with my own alternative code

(which still relies on tricky bit manipulation, see bits.stephan-brumme.com).

# My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too.

#include <iostream>
#include <vector>
// represent digits as bitmasks, lit segments are 1

const unsigned char Segments[10] =
{
// assume the following numbering scheme for the seven segments:
// 000
// 1 2
// 1 2
// 333
// 4 5
// 4 5
// 666
(1 << 0) | (1 << 1) | (1 << 2) | (0 << 3) | (1 << 4) | (1 << 5) | (1 << 6), // digit 0 = 0x77
(0 << 0) | (0 << 1) | (1 << 2) | (0 << 3) | (0 << 4) | (1 << 5) | (0 << 6), // digit 1 = 0x24
(1 << 0) | (0 << 1) | (1 << 2) | (1 << 3) | (1 << 4) | (0 << 5) | (1 << 6), // digit 2 = 0x5d
(1 << 0) | (0 << 1) | (1 << 2) | (1 << 3) | (0 << 4) | (1 << 5) | (1 << 6), // digit 3 = 0x6d
(0 << 0) | (1 << 1) | (1 << 2) | (1 << 3) | (0 << 4) | (1 << 5) | (0 << 6), // digit 4 = 0x2e
(1 << 0) | (1 << 1) | (0 << 2) | (1 << 3) | (0 << 4) | (1 << 5) | (1 << 6), // digit 5 = 0x6b
(1 << 0) | (1 << 1) | (0 << 2) | (1 << 3) | (1 << 4) | (1 << 5) | (1 << 6), // digit 6 = 0x3b
(1 << 0) | (1 << 1) | (1 << 2) | (0 << 3) | (0 << 4) | (1 << 5) | (0 << 6), // digit 7 = 0x27
(1 << 0) | (1 << 1) | (1 << 2) | (1 << 3) | (1 << 4) | (1 << 5) | (1 << 6), // digit 8 = 0x7f
(1 << 0) | (1 << 1) | (1 << 2) | (1 << 3) | (0 << 4) | (1 << 5) | (1 << 6) // digit 9 = 0x6f
};
// ---------- standard prime sieve from my toolbox ----------

// odd prime numbers are marked as "true" in a bitvector

std::vector<bool> sieve;
// return true, if x is a prime number

bool isPrime(unsigned int x)
{
// handle even numbers
if ((x & 1) == 0)
return x == 2;
// lookup for odd numbers
return sieve[x >> 1];
}
// find all prime numbers from 2 to size

void fillSieve(unsigned int size)
{
// store only odd numbers
const unsigned int half = (size >> 1) + 1;
// allocate memory
sieve.resize(half, true);
// 1 is not a prime number
sieve[0] = false;
// process all relevant prime factors
for (unsigned int i = 1; 2 * i*i < half; i++)
// do we have a prime factor ?
if (sieve[i])
{
// mark all its multiples as false
unsigned int current = 3 * i + 1;
while (current < half)
{
sieve[current] = false;
current += 2 * i + 1;
}
}
}
// ---------- problem solution ----------

// return sum of all digits

unsigned int digitSum(unsigned int x)
{
unsigned int result = 0;
while (x > 0)
{
result += x % 10;
x /= 10;
}
return result;
}
// convert number to its bitmasks, 8 bits per digit => x must not exceed 99999999

unsigned long long getSegments(unsigned int x)
{
unsigned long long result = 0;
unsigned int shift = 0;
while (x > 0)
{
unsigned long long current = Segments[x % 10];
result |= current << shift;
x /= 10;
shift += 8;
}
return result;
}
// count bits which are set to 1

unsigned int popcnt(unsigned long long x)
{
#ifdef __GNUC__
return __builtin_popcount(x);
#endif
unsigned int result = 0;
while (x > 0)
{
// clear right-most bit
x &= x - 1;
result++;
}
return result;
}
// process all steps with strategy of Sam, return number of transitions

unsigned int sam(unsigned int x)
{
// memoize (runs quite fast without, too)
static std::vector<unsigned int> cache(100, 0);
if (x < cache.size() && cache[x] > 0)
return cache[x];
// display was clear
auto segments = getSegments(x);
// switch (intermediate) number on and off
auto result = 2 * popcnt(segments);
// need further iterations ?
if (x > 9)
result += sam(digitSum(x));
if (x < cache.size())
cache[x] = result;
return result;
}
// process all steps with strategy of Max, return number of transitions

unsigned int max(unsigned int x, unsigned long long previousSegments = 0)
{
auto segments = getSegments(x);
auto transitions = segments ^ previousSegments;
auto result = popcnt(transitions);
if (x > 9)
// morph to next number
result += max(digitSum(x), segments);
else
// fade out
result += popcnt(segments);
return result;
}
int main()
{
// search range
unsigned int from = 10000000;
unsigned int to = 20000000;
std::cin >> from >> to;
// for live input only
if (from > to)
return 1;
// generate prime sieve
fillSieve(to);
// count all digits of Sam and Max
unsigned long long sumSam = 0;
unsigned long long sumMax = 0;
for (auto i = (from | 1); i <= to; i += 2) // odd numbers only
{
if (!isPrime(i))
continue;
sumSam += sam(i);
sumMax += max(i);
}
// Sam >= Max, display the difference
std::cout << sumSam - sumMax << std::endl;
return 0;
}

This solution contains 27 empty lines, 39 comments and 4 preprocessor commands.

# Interactive test

You can submit your own input to my program and it will be instantly processed at my server:

This is equivalent to`echo "137 137" | ./315`

Output:

*Note:* the original problem's input `10000000 20000000`

__cannot__ be entered

because just copying results is a soft skill reserved for idiots.

*(this interactive test is still under development, computations will be aborted after one second)*

# Benchmark

The correct solution to the original Project Euler problem was found in 0.13 seconds on a Intel® Core™ i7-2600K CPU @ 3.40GHz.

Peak memory usage was about 3 MByte.

(compiled for x86_64 / Linux, GCC flags: `-O3 -march=native -fno-exceptions -fno-rtti -std=c++11 -DORIGINAL`

)

See here for a comparison of all solutions.

**Note:** interactive tests run on a weaker (=slower) computer. Some interactive tests are compiled without `-DORIGINAL`

.

# Changelog

July 29, 2017 submitted solution

July 29, 2017 added comments

# Difficulty

Project Euler ranks this problem at **20%** (out of 100%).

# Links

projecteuler.net/thread=315 - **the** best forum on the subject (*note:* you have to submit the correct solution first)

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# Copyright

I hope you enjoy my code and learn something - or give me feedback how I can improve my solutions.

All of my solutions can be used for any purpose and I am in no way liable for any damages caused.

You can even remove my name and claim it's yours. But then you shall burn in hell.

The problems and most of the problems' images were created by Project Euler.

Thanks for all their endless effort.

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